Tied dice. II. Some asymptotic results

Tied dice. II. Some asymptotic results Cooley, C.; Ella, W.; Follett, M.; Gilson, E.; Traldi, Lorenzo A dice family D(n,a,b,s) includes all lists (x1,…,xn) of integers with n≥1, a≤x1≤⋯≤xn≤b and Σxi=s. Given two dice X and Y we compare the number of pairs (i,j) with xiyj. If the second number is larger than X is stronger than Y, and if the two numbers are equal then X and Y are tied. In previous work it has been observed that the density of ties in D(n,a,b,s) is generally lower than one might expect. In this note we provide more information about this observation by calculating the asymptotic proportion of ties in certain kinds of dice families. Many other properties of dice families remain to be determined.